Defining Simple Harmonic Motion and Restoring Force

Simple harmonic motion (SHM) is periodic motion in which the acceleration is proportional to the displacement from the equilibrium position and directed toward it, a = −ω²x, so the displacement varies sinusoidally with time, x = A cos(ωt + φ), with period T = 2π ÷ ω. In brief: the restoring force is proportional to the displacement, F = −kx (Hooke’s law). The amplitude A is the maximum displacement, the maximum speed is v_max = Aω (at equilibrium) and the maximum acceleration is a_max = Aω² (at the extremes). The period is independent of the amplitude.

The motion of a mass on a spring, a pendulum swinging through small angles, the vibrations of atoms in a solid and the oscillation of a guitar string are all approximately SHM. This guide defines SHM and the restoring force, gives the equations of displacement, velocity and acceleration, explains graphs and phase, compares the quantities at different points of the cycle in a table, works through numerical examples and corrects common mistakes.

What is simple harmonic motion?

An oscillation is a repeated back-and-forth motion about an equilibrium position. SHM is the simplest and most important type. It occurs when the net force on the object is a restoring force that is proportional to the displacement from equilibrium and directed toward it: F = −kx. By Newton’s second law, ma = −kx, so a = −(k ÷ m)x = −ω²x, where ω = √(k ÷ m). Two conditions define SHM: the acceleration is proportional to the displacement, and it is in the opposite direction (toward the equilibrium point). Terms: the amplitude A (maximum displacement), the period T (time for one complete oscillation), the frequency f = 1 ÷ T, and the angular frequency ω = 2πf = 2π ÷ T.

What are the equations of SHM?

The solution of a = −ω²x is a sinusoidal function. If the motion starts at the maximum displacement at t = 0:

• displacement: x = A cos ωt (or x = A sin ωt if it starts at equilibrium); in general x = A cos(ωt + φ), where φ is the phase constant;
• velocity: v = −Aω sin ωt, with the maximum Aω, and in terms of position, v = ±ω√(A² − x²);
• acceleration: a = −Aω² cos ωt = −ω²x, with the maximum Aω².

SHM is the projection onto a diameter of uniform circular motion: a point moving around a circle of radius A at angular speed ω has a shadow that moves in SHM. The period T = 2π ÷ ω does not depend on the amplitude (isochronism), which is why pendulum clocks keep time as the swings die down, provided the amplitude is small.

What do the graphs look like?

The displacement-time graph is a cosine (or sine) curve. The velocity-time graph is the same shape shifted by a quarter of a period (π/2 in phase): velocity is zero at the extremes and maximum at equilibrium. The acceleration-time graph is the negative of the displacement graph (in antiphase). The slope of the displacement graph gives the velocity; the slope of the velocity graph gives the acceleration. A graph of acceleration against displacement is a straight line through the origin with a negative gradient −ω², which is a test for SHM.

Quantities during the cycle

Worked example: a spring oscillator

A mass on a spring oscillates with amplitude 5.0 cm (0.050 m) and frequency 2.0 Hz. It is released from the maximum displacement at t = 0.

1. Angular frequency: ω = 2πf = 2π × 2.0 = 12.6 rad/s. Period T = 1 ÷ 2.0 = 0.50 s.
2. Maximum speed: v_max = Aω = 0.050 × 12.6 = 0.63 m/s, at the equilibrium position.
3. Maximum acceleration: a_max = Aω² = 0.050 × 158 = 7.9 m/s², at the extremes.
4. Displacement at t = 0.10 s: x = A cos ωt = 0.050 × cos(12.57 × 0.10) = 0.050 × cos(1.257 rad) = 0.050 × 0.309 = 0.0155 m = 1.5 cm.
5. Speed at x = 3.0 cm: v = ω√(A² − x²) = 12.6 × √(0.0025 − 0.0009) = 12.6 × 0.040 = 0.50 m/s.
6. Check of the condition: a = −ω²x = −158 × 0.030 = −4.7 m/s² at x = +3.0 cm: the acceleration is toward the center and proportional to the displacement.

How was SHM studied?

Galileo observed the isochronism of a swinging lamp in the cathedral of Pisa in 1583 (according to a story told by his biographer). Robert Hooke published his law of springs in 1678, and Christiaan Huygens’ pendulum clocks (1656) used the property. Joseph Fourier showed in 1822 that any periodic motion can be written as a sum of sinusoidal motions, which makes SHM the building block of wave and vibration analysis.

Which mistakes are common?

Mistake 1: Thinking that the acceleration is constant

It changes continuously, with a = −ω²x. The constant-acceleration equations do not apply.

Mistake 2: Saying that the speed is greatest at the extremes

The speed is zero at the extremes and greatest at the equilibrium position.

Mistake 3: Using degrees in the calculator

ωt must be in radians in x = A cos ωt.

Mistake 4: Believing that the period depends on the amplitude

For SHM, it does not. For a pendulum, this holds only for small angles.